Here \(C^+(\cdot , \cdot )\)  and \(IG(\cdot , \cdot )\)  respectively denote the half Cauchy density and Inverse Gamma density.
The level of sparsity induced in the model is controlled by the global scale parameter \(\tau \) , and choosing the same is a non-trivial problem. Using a plug-in estimate for \(\tau \)  based on empirical Bayes method suffers from a potential danger of resulting in a degenerate solution resulting in a heavily sparse model. There are several works [1]}, [2]}, [3]} that suggest effective methods regarding the choice of \(\tau \) . In this paper, we have proposed to take a fully Bayesian approach as suggested in [1]}, [2]} and use a half-Cauchy prior.
